This paper describes a method for developing the capability of the inverse problem of eigenstrain in allowing for the determination of the spatially varying multiaxial residual stresses at the microscale level, based on the assumption that the residual stresses are due to an inelastic misfit strain (eigenstrain). Carbide particles in superalloy MAR-M-002 subjected to thermally induced strains due to mismatched thermal expansion coefficients have been studied in this work. We take as a starting point the residual elastic strain tensor measured by cross-correlation-based analysis of electron back-scatter diffraction (EBSD) patterns obtained from the surface of a sample sectioned through the carbide particle. From this we calculate the multiaxial residual stresses that existed in the sample prior to sectioning. In addition, a finite-element simulation of the inverse problem of eigenstrain has been carried out to validate numerically the inverse method.